The Jamming Energy Landscape is Hierarchical and Ultrametric

Recent infinite dimensional mean field results have shown that the energy landscape of glasses forms a hierarchically structured ultrametric space. We directly explore this prediction in low dimensional finite size systems. Starting in an energy minima of an over-jammed system, we apply a small random perturbation to the positions of our particles and re-minimize in order to find new minima. By enumerating these minima and calculating the distance between their force networks, we produce a small correlated portion of the overlap matrix. This matrix exhibits a clear hierarchical structure and shows the signature of an ultrametric space. That such a hierarchy exists for the jamming energy landscape provides direct evidence for the existence of a marginal phase created by the Gardner transition.